What are the advantages of complex-valued neural networks?
According to this paper, complex-valued ANNs (C-ANNs) can solve problems such as XOR and symmetry detection with a smaller number of layers than real ANNs (for both of these a 2 layer C-ANN suffices, whereas a 3-layer R-ANN is required).

I believe that it is still an open question as to how useful this result is in practice (e.g. whether it actually makes finding the right topology easier), so at present the key practical advantage of C-ANNs is when they are a closer model for the problem domain.

Application areas are then where complex values arise naturally, e.g. in optics, signal processing/FFT or electrical engineering.